Introduction to Engineering Calculations

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Introduction to Engineering Calculations

• Chapter 2

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Whats in this chapter?

• Conversion factors
• Units
• Unit Conversions
• Significant figures
• Dimensional analysis
• Graphical analysis of data

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Introduction

• Describe the basic techniques for the handling of
  units and dimensions in calculations.
• Describe the basic techniques for expressing the
  values of process variables and for setting up
  and solving equations that relate these
  variables.
• Develop an ability to analyze and work
  engineering problems by practice.

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Units and Dimensions

• Convert one set of units in a function or
  equation into another equivalent set for mass,
  length, area, volume, time, energy and force
• Specify the basic and derived units in the SI and
  American engineering system for mass, length,
  volume, density, time, and their equivalence.
• Explain the difference between weight and mass
• Apply the concepts of dimensional consistency to
  determine the units of any term in a function

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Units and Dimensions

• Dimensions are properties that can be measured
  such as length, time, mass, temperature, or
  calculated by multiplying or dividing other
  dimensions, such as velocity (length/time)
• Units are means of expressing the dimensions such
  as feet or meter for length, hours/seconds for
  time.
• Every valid equation must be dimensionally
  homogeneous that is, all additive terms on both
  sides of the equation must have the same unit

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CONVERSION OF UNITS

• A measured quantity can be expressed in terms of
  any units having the appropriate dimension
• To convert a quantity expressed in terms of one
  unit to equivalent in terms of another unit,
  multiply the given quantity by the conversion
  factor

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Example

• If you are given a quantity having a compound
  unit, and wish to convert to another set of units
  for instance, to convert an acceleration of 1
  in/s2 to miles/year2, set a dimensional equation.
  

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SYSTEMS OF UNITS

• Components of a system of units
• Base units - units for the dimensions of mass,
  length, time, temperature, electrical current,
  and light intensity.
• Derived units - units that are obtained in one
  or two ways
• By multiplying and dividing base units also
  referred to as compound units
• Example ft/min (velocity), cm2(area), kg.m/s2
  (force)
• SI and American engineering system units.

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Common Systems of Units
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Common Systems of Units
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Force and Weight

• Be sure you understand the difference between lbf
  and lbm
• Be sure you understand the difference between the
  physical constant g, and the conversion factor gc.

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FORCE, WEIGHT AND MASS

• Force is proportional to product of mass and
  acceleration and is defined using derived units
  to equal the natural units
• 1 Newton (N) 1 kg.m/s2
• 1 dyne 1 g.cm/s2
• 1 Ibf 32.174 Ibm.ft/s2
• Weight of an object is force exerted on the
  object by gravitational attraction of the earth
  i.e. force of gravity, g.
• To convert a force from a derived force unit to a
  natural unit, a conversion factor, gc must be
  used.
• A ratio of gravitational acceleration, g to gc
  may be used for most conversions between mass and
  weight.

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FORCE, WEIGHT AND MASS
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Example

• The density of a fluid is given by the empirical
  equation
• r 1.13 exp(1.2 x 10-10 P)
• Where r density in g/cm3
• P pressure in N/m2
• a) What are the units of 1.13 and 1.2 x 10-10?
• b) Derive the formula for r(Ibm/ft3) as a
  function of P (Ibf/in2)
• A column of mercury is 3 mm in diameter x 72 cm
  high. If the density of mercury is 13.6 g/cm3,
  what is its weight in N. What is its weight in
  Ibf? What is its mass in Ibm?

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Example

• The Reynolds number is the dimensionless quantity
  that occurs frequently in the analysis of the
  flow of fluids. For flow in pipes it is defined
  as DVr/m, where D is the pipe diameter, V is the
  fluid velocity, r is the fluid density, and m is
  the fluid viscosity. For a particular system
  having D 4.0 cm, V 10.0 ft/s, r 0.700
  g/cm3, and m 0.18 centipoise (cP) (where 1 cP
  6.72 x 10-4 Ibm/ft.s). Calculate the Reynolds
  number.

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Numerical Calculations and Estimation

• Scientific Notation
• Engineering Notation
• Significant Figures
• Precision
• Precision vs accuracy

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Validating Results

• Back substitution
• Plug your answer back in and see if it works
• Order of magnitude estimation
• Round off the inputs, and check to see if your
  answer is the right order of magnitude
• Reasonableness does it make sense
• If you get a negative temp in K, you probably
  have done something wrong

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Statistical Calculations

• Mean
• Range
• Sample Variance
• Sample Standard Deviation

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Sample Means
Most measured amounts are means
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All means are not created equal
Consider these two sets of data
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Range
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Sample Variance
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Standard Deviation
Your calculator will find all of these
statistical quantities for you
Spreadsheets also have built in statistical
functions
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Standard Deviation

• For typical random variables, roughly 2/3 of all
  measured values fall within one standard
  deviation of the mean
• About 95 fall inside 2 standard deviations
• About 99 fall within 3 standard deviations

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Data Representation

• Collected data has scatter
• Calibration

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Two Point Linear Interpolation

• Dont get confused by the funky equation

This works if you have a lot of tabulated data
for your linear interpolation
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Fitting a Straight Line

• A more general and more compact way to represent
  how one variable depends on another is with an
  equation
• Lets look at straight lines first
• yaxb

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Example 2.7-1
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(No Transcript)
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What if the relationship between x and y isnt a
straight line?

• Plot it so that it is a straight line
• Why?
• Look at page 25

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Plot y vs x2
Plot y2 vs 1/x
Lets try Example 2.7-2 Use Excel as our graphing
tool
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Common non-linear functions

• Exponential
• Power Law

If you plot the ln(y) vs x, you get a straight
line
If you plot the ln(y) vs ln(x) you get a straight
line
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Use Excel to make these plots

• Use the trendline to find the equation of the
  best fit line

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Homework

• Chapter 2
• 2.6
• 2.10
• 2.18
• 2.22
• 2.23
• 2.32